How to get rid of these displacement artifacts?
If I understand you correct, it is the sudden drop in the terrain and the jagged edge that you want to avoid. You must smoothe the edges of the heightmap. 

The transparent portion of your texture is treated as white. So it goes quite abruptly from dark to white.

One way to do this is in photoshop.


Duplicate the layer.
Smooth the bottom layer. (the smaller blur radius the better result)
Merge the two layers.
Repeat this until the entire image is filled.


It is handy to make this an action (a macro or a script) in photoshop. It can easily be recorded.



Halfway step:


Final result: